International Journal of Computer Mathematics. Fuzzy congruence relations on nd-groupoids. Journal: International Journal of Computer Mathematics
|
|
- Eustace Rodgers
- 5 years ago
- Views:
Transcription
1 Fuzzy congruence relations on nd-groupoids Journal: Manuscript ID: GCOM-00-0-A.R Manuscript Type: Original Article Date Submitted by the Author: n/a Complete List of Authors: Ojeda-Aciego, Manuel; Universidad Malaga Cordero, Pablo; Universidad de Málaga de las Peñas, Inmaculada; Universidad de Málaga Gutiérrez, Gloria; Universidad de Málaga Martínez, Javier; Universidad de Málaga Keywords: fuzzy congruence relation, hypergroupoid, nd-groupoid, nondeterministic operations, complete lattice, 0A, 0B0, 0N0, E
2 December, 00 : FuzzyCong(IJCM Page of 0 0 Vol. 00, No. 00, Month 00x, RESEARCH ARTICLE Fuzzy congruence relations on nd-groupoids P. Cordero, I. de las Peñas, G. Gutiérrez, J. Martínez and M. Ojeda-Aciego Department of Applied Mathematics. University of Málaga. Spain (Received 00 Month 00x; in final form 00 Month 00x In this work we introduce the notion of fuzzy congruence relation on an nd-groupoid and study conditions on the nd-groupoid which guarantee a complete lattice structure on the set of fuzzy congruence relations. The study of these conditions allowed to construct a counterexample to the statement that the set of fuzzy congruences on a hypergroupoid is a complete lattice. Keywords: fuzzy congruence relation, nd-groupoid, hypergroupoid MSC codes: 0A, 0B0, 0N0, E. Introduction The systematic generalization of crisp concepts to the fuzzy case has proven to be an important theoretical tool for the development of new methods of reasoning under uncertainty, imprecision and lack of information. Regarding the generalization level, it is important to note that the definition of fuzzy sets originally presented as mappings with codomain [0, ], was soon replaced by more general structures, for instance a complete lattice, as in the L-fuzzy sets introduced by Goguen []. This paper continues previous work [, ] which is aimed at investigating L-fuzzy sets where L has the structure of a multilattice, a structure introduced in [] and later recovered for use in other contexts, both theoretical and applied [0, ]. Roughly speaking, a multilattice is an algebraic structure in which the restrictions imposed on a lattice, namely, the existence of least elements in the sets of upper bounds and greatest elements in the sets of lower bounds are relaxed to the existence of minimals and maximals, respectively, in the corresponding sets of bounds. Attending to this informal description, the main difference that one notices when working with multilattices is that the operators which compute suprema and infima are no longer single-valued, since there may be several multi-suprema or multi-infima, or may be none, see Figure. This immediately leads to the theory of hyperstructures, that is, algebras whose operations are set-valued. If A is a non-empty set and H is a family of set-valued operations on A, the ordered pair (A, H is called a hyperalgebra (or multialgebra, or polyalgebra. The study of hyperalgebras was originated in when Marty introduced the so-called hypergroups in []. Since then, a number of papers have been published on this topic, focussing essentially on special types of hyperalgebras (such as hypergroups, hyperrings, hyperfields, vector hyperspaces, boolean hyperalgebras,... Corresponding author. aciego@uma.es ISSN: 000- print/issn 0-0 online c 00x Taylor & Francis DOI: 0.00/000YYxxxxxxxx
3 December, 00 : FuzzyCong(IJCM Fuzzy congruence relations on nd-groupoids Page of 0 0 c d a b 0 Figure. A complete multilattice which is not a lattice. and guided, sometimes by purely theoretical motivations and sometimes because of their applications in other areas. In this paper, we will focus on the most general hyperstructures, namely hypergroupoids and nd-groupoids. Our interest in these structures arises from the fact that, in a multilattice, the operators which compute the multi-suprema and multiinfima provide precisely the structure of nd-groupoids or, if we have for granted that at least a multi-supremum always exists, a hypergroupoid. Actually, some of the results will be stated just in terms of multisemilattices. Several papers have investigated the structure of the set of fuzzy congruences on different algebraic structures [,,,, ]; and in [, ] we initiated our research in this direction. Specifically, we focused on the theory of (crisp congruences on a multilattice and on an nd-groupoid, as a necessary step before studying the fuzzy congruences on multilattices and the multilattice-based generalization of the concept of L-fuzzy congruence. In this paper, we study the notion of fuzzy congruence relation on nd-groupoids. The fact that the structure of nd-groupoid is simpler than that of a multilattice does not necessarily mean that the theory is simpler as well. We will show that, in general, the set of fuzzy congruences on an nd-groupoid is not a lattice unless we assume some extra properties. The structure of the paper is as follows: in Section, the preliminary definitions and concepts related to nd-groupoids and fuzzy congruences are introduced; then, in Section, based on the definition of fuzzy congruence on a hypergroupoid given in [], we introduce our generalization to the context of nd-groupoids, and prove that a fuzzy relation ρ is a fuzzy congruence relation on an nd-groupoid A if and only if its power extension ρ is a fuzzy congruence relation on the powerset groupoid A. Section is devoted to the study of the lattice structure of fuzzy congruence relations, the two main results being that, ( contrariwise to what is stated in [], Theorem., the set of fuzzy congruences on A, FCon(A, is not always a lattice, and ( sufficient conditions to prove that FCon(A is a complete lattice. In the final section, we draw some conclusions and present prospects of future work.. Preliminaries We can find in the literature the definition of a hypergroupoid as a non-empty set endowed with a hyperoperation : A A A { }. However, we are interested in a generalization of hypergroupoid that we will call non-deterministic groupoid (nd-groupoid, for short which also considers the empty set as possible image of the hyperoperation. Definition. An nd-groupoid (A, is defined by an nd-operation : A A A on a non-empty set A. The induced power groupoid is defined as ( A, where
4 December, 00 : FuzzyCong(IJCM Page of I.P. Cabrera, P. Cordero, G. Gutiérrez, J. Martínez, M. Ojeda-Aciego 0 0 the operation is given by X Y = {x y x X,y Y } for all X,Y A. Notice that the definition allows the assignment of the empty set to a pair of elements, that is a b =. This mere fact, albeit simple, represents an important difference with hypergroupoids, as it will be explained later. The following notational conventions will be used hereafter: Multiplicative notation; thus, the symbol of the nd-operation will be omitted. If a A and X A, we will denote ax = {ax x X} and Xa = {xa x X}. In particular, a = a =. When the result of the nd-operation is a singleton, we will often omit the braces. As stated in the introduction, our interest in extending the concept of hypergroupoid is justified by the algebraic characterization of multilattices and multisemilattices, since the operators for multi-suprema and multi-infima are both examples of nd-operators. With this idea in mind, we introduce below the extension to the framework of nd-groupoids of some well-known properties. Assume that (A, is an nd-groupoid: Idempotency: aa = a for all a A. Commutativity: ab = ba for all a, b A. Left m-associativity: if ab = b, then (abc a(bc, for all a,b,c A. Right m-associativity: if bc = c, then a(bc (abc, for all a,b,c A. m-associativity: if it is left and right m-associative. Note that the prefix m- has its origin in the concept of multilattice. We will focus our interest on the binary relation usually named natural ordering, which is defined by a b if and only if ab = b Although, in general, this relation is not an ordering, the properties above guarantee that the relation just defined is an ordering. Specifically, it is reflexive if the nd-groupoid is idempotent, the relation is antisymmetric if the nd-groupoid is commutative and, finally, it is transitive if the nd-groupoid is m-associative. The two following properties of nd-groupoids, named comparability properties, have an important role in multilattice theory: C : c ab implies that a c and b c. C : c,d ab and c d imply that c = d. Similarly to lattice theory, we can define algebraically the concept of multisemilattice as an nd-groupoid that satisfies idempotency, commutativity, m-associativity and comparability laws. The ordered and the algebraic definitions of multisemilattice can be proved to be equivalent simply by considering a b = multisup{a,b} and being the natural ordering (see [], Theorem.. Since our aim is to extend the results about fuzzy congruences to nd-groupoids and multisemilattices, let us recall some notions about the concepts that we will generalize. Definition. (Zadeh, [] Let A be a non-empty set. A fuzzy relation ρ on A is a fuzzy subset of A A (i.e. ρ is a function from A A to [0,]. ρ is reflexive in A if ρ(x,x = for all x A, ρ is symmetric in A if ρ(x,y = ρ(y,x for all x, y A, finally, ρ is transitive if supmin {ρ(x,z,ρ(z,y} ρ(x,y for all x,y A z A
5 December, 00 : FuzzyCong(IJCM Fuzzy congruence relations on nd-groupoids Page of 0 0 A fuzzy equivalence relation is a reflexive, symmetric and transitive fuzzy relation. Since a fuzzy relation in a non-empty set A is a fuzzy subset of A A, we can define the inclusion, union and intersection of fuzzy relations as follows: ρ σ if ρ(x,y σ(x,y for all x,y A, ρ i (x,y = supρ i (x,y for all x,y A and i Λ i Λ ρ i (x,y = inf ρ i(x,y for all x,y A. i Λ i Λ Definition. (Zadeh, [] Let A be a non-empty set and ρ and σ two fuzzy relations in A. Then we define the sup-min composition of ρ and σ as: (ρ σ(x,y = supmin {ρ(x,z,σ(z,y} for all x,y A z A It is easy to prove that the sup-min composition of two fuzzy relations is associative. Moreover, a fuzzy relation ρ is transitive on A if ρ ρ ρ. Let FEq(A be the set of fuzzy equivalence relations on a non-empty set A. Murali proved in [] that (FEq(A, is a complete lattice where the meet is the intersection and the join is the transitive closure of the union. Finally, let us introduce the definition of fuzzy congruence on a groupoid. Definition. Let ρ be a fuzzy relation on a groupoid (G, ; we say that ρ is right compatible with the operation if ρ(ac,bc ρ(a,b for all a,b,c G; similarly, ρ is said to be left compatible if ρ(ca,cb ρ(a,b for all a,b,c, G. A fuzzy congruence on G is a left and right compatible fuzzy equivalence relation.. Fuzzy congruence relations on nd-groupoids Regarding the extension of the definition of fuzzy congruence to the nondeterministic case, the following definition of compatibility, in the case of an underlying hypergroupoid, was introduced by Bakhshi and Borzooei in []. Definition. Let (A, be an nd-groupoid. Then a fuzzy relation ρ on A is said to be right (left compatible if for all x ac (x ca there exists y bc (y cb and for all y bc (y cb there exists x ac (x ca such that ρ(x,y ρ(a,b, for all a,b,c A and compatible if it is both fuzzy right and left compatible. This definition explicitly uses the fact that the images of the hyperoperator are non-empty. Thus, we propose an alternative definition which generalizes the previous one and adequately handles the empty images. As a previous step to the consideration of fuzzy congruence relations on an ndgroupoid, let us note that it is possible to extend any fuzzy relation on a set A to its powerset A ; this construction leads to the definition of an operator from the set FR(A of fuzzy relations on A to the set FR( A of fuzzy relations on A. Namely, given a fuzzy relation ρ : A A [0,], its power extension is a fuzzy relation ρ : A A [0,] defined by ρ(x,y = ρ(x, y y Y x X ρ(x, y where and denotes the supremum and the infimum in the unit interval, respectively.
6 December, 00 : FuzzyCong(IJCM Page of I.P. Cabrera, P. Cordero, G. Gutiérrez, J. Martínez, M. Ojeda-Aciego 0 0 Notice that ρ(,x = ρ(x, = 0, for all non-empty X A, ρ(, = and ρ({a}, {b} = ρ(a,b, for all a,b A. With this power extension of a fuzzy relation, the definition of fuzzy congruence relation on an nd-groupoid (A, follows exactly the one for the deterministic case: a fuzzy equivalence relation that satisfies ρ(ac,bc ρ(a,b and ρ(ca,cb ρ(a,b, for all a,b,c A ( It is easy to check that a fuzzy relation that is compatible with (in the sense of Definition. satisfies Condition ( but, in general, both concepts are not equivalent as the following example shows: Example. Let A = [0, ] be the hypergroupoid endowed with the hyperoperation a b := (0, and consider the fuzzy relation ρ(a, b = ab. Observe that ρ(a c,b c = x (0, y (0, = x (0, ( xy y (0, y (0, x (0, = ρ(a,b ( xy = for all a,b,c A. However, for all x 0 c and y b c, we have ρ(x,y < ρ(0, b = because otherwise, we would have either x = 0 or y = 0 contradicting that x,y (0,. Thus, ρ is not compatible with the hyperoperation. Once we have introduced the power extension of a fuzzy relation, in order to use the above condition to define the concept of fuzzy congruence relation, we study the behaviour of the operator wrt the properties of reflexivity, simmetry and transitivity. Theorem. Let ρ be a fuzzy relation in a non-empty set A and let ρ be its power extension as defined above. ρ is a fuzzy equivalence relation if and only if so is ρ. Proof For the cases of reflexivity and antisymmetry it is just a matter of routine calculation. Let us concentrate on transitivity. Under the assumption that ρ is transitive on A, it is sufficient to prove that, for all X,Y,Z, A, we have that ρ(x,y ρ(y,z ρ(x,z, in order to ensure that ρ is transitive. ρ(x,y ρ(y,z = = ρ(x,y = y Y x X ρ(x,y y Y z Z ρ(x,y ρ(y,z ρ(y, z ] ρ(y, z y Y z Z z Z y Y z Z y Y ρ(y,z y Y x X ] ρ(x, y Now, by idempotency and distributivity, we have that ρ(x,y ρ(y,z equals (ρ(x,y y Y z Z ] ρ(y,z z Z y Y (ρ(y,z y Y x X ] ρ(x,y
7 December, 00 : FuzzyCong(IJCM 0 0 Fuzzy congruence relations on nd-groupoids As y Y z Z ρ(y,z z Z ρ(y,z, for all y Y, we have that ρ(x,y ρ(y,z (ρ(x,y = z Z = x X z Z y Y z Z ] ρ(y, z ( ] ρ(x,y ρ(y,z ( ] ρ(x,y ρ(y,z z Z y Y (ρ(y,z z Z y Y x X x X z Z Since ρ is transitive, ρ(x,z (ρ(x, y ρ(y, z, and so y Y ρ(x,y ρ(y,z x X z Z ] ρ(x, z z Z x X and as this is true for all Y A (including Y =, we have that and so ρ is transitive. Conversely, if ρ is transitive, then ρ(a,b = ρ({a}, {b} {z} A Y A ( ρ(x,y ρ(y,z ρ(x,z Z A ( ρ({a},z ρ(z, {b} ] ρ(x, y ( ] ρ(y,z ρ(x,y ( ] ρ(x,y ρ(y,z ] ρ(x, z = ρ(x,z ( ( ρ({a}, {z} ρ({z}, {b} = ρ(a,z ρ(z,b Summarizing the previous considerations we can state the following definition and theorem. Definition. A fuzzy equivalence relation ρ on an nd-groupoid (A, is said to be a right (resp. left congruence relation if ρ(ac,bc ρ(a,b (resp. ρ(ca,cb ρ(a, b for all a, b, c A. A fuzzy relation is said to be a congruence relation if it is a left and right congruence relation. Notice that, henceforth, in order to avoid repetitions, we will only concentrate on the right versions of properties. Theorem. Let ρ be a fuzzy relation on an nd-groupoid (A,. Then, ρ is a fuzzy congruence relation if and only if ρ is a fuzzy congruence relation in the induced power groupoid ( A,. Proof By Theorem., we only need to prove the compatibility with the ndoperation. If ρ is a congruence in ( A, then, for all a,b,c A, z A ρ(ac,bc = ρ({a}{c}, {b}{c} ρ({a}, {b} = ρ(a,b Page of
8 December, 00 : FuzzyCong(IJCM Page of I.P. Cabrera, P. Cordero, G. Gutiérrez, J. Martínez, M. Ojeda-Aciego 0 0 Conversely, for all X,Y,Z A, ρ(xz,y Z = xz XZ yz Y Z xz XZ y Y ρ(xz,yz ρ(xz, yz ρ(x, y y Y x X yz Y Z xz XZ yz Y Z x X ρ(xz, yz ρ(xz,yz ρ(x, y = ρ(x,y The sup property, introduced in Definition. below, guarantees the equivalence between our definition of fuzzy congruence relation and the one given in []. Definition. Let A be a non-empty set and ρ a fuzzy relation on A. We say that ρ satisfies the right (resp. left sup property if for all a A and for all non-empty X A, there exists y 0 X (resp. x 0 X such that sup y X ρ(a,y = ρ(a,y 0 (resp. sup x X ρ(x,a = ρ(x 0,a. Lemma. Let ρ be a fuzzy equivalence relation on an nd-groupoid (A, which satisfies sup property. Then, ρ is a fuzzy congruence relation if and only if ρ is compatible with the nd-operation. Proof Let us suppose that ρ is compatible. Let x ac, then there exists y bc such that ρ(x,y ρ(a,b. So ρ(x,y ρ(a,b and ρ(x,y ρ(a,b. Analogously y bc x ac y bc ρ(x, y ρ(a, b therefore ρ(ac,bc ρ(a,b x ac y bc Notice that the sup property is not required in this implication. For the converse, we only check the first condition of Definition. because the other ones follows the same scheme. If ρ is a fuzzy congruence relation, then ρ(ac,bc ρ(a,b. In particular ρ(x,y ρ(a,b. By the sup property, for x ac y bc all x ac there exists y 0 bc such that ρ(x,y = ρ(x,y 0. Since ρ(x,y 0 ρ(x,y 0, we obtain ρ(a,b ρ(x,y 0 y bc. On the lattice structure of fuzzy congruence relations In the previous section, we introduced the map : FR(A FR( A and proved that ρ FR(A is a fuzzy equivalence relation if and only if ρ is a fuzzy equivalence relation. Let us now consider this map on FCon(A, the subset of FEq(A consisting of the fuzzy congruence relations. First, notice that Theorem. guarantees that : FCon(A FCon( A is well defined. In the crisp case, Murali proved in [] that the set of fuzzy congruence relations on a groupoid X is a complete sublattice of the set of all fuzzy equivalence relations. This result might suggest that the lattice structure of FCon( A can be projected on FCon(A, via the map. However, although ρ is injective, since ρ({a}, {b} = x ac
9 December, 00 : FuzzyCong(IJCM Fuzzy congruence relations on nd-groupoids Page of 0 0 ρ(a,b, for all a,b A, it is not surjective. If it were surjective, then for all Θ FCon( A the following equality would hold Θ(X,Y = Θ({x}, {y} y Y x X Θ({x}, {y} but, in general, this is not the case, as the following example shows. Example. Let (A, be the nd-groupoid with A = {a,b} and x y = {a}, for all x,y A. Let Θ be the reflexive and symmetric fuzzy relation on A given by Θ({a}, {b} = ;Θ({a},A = Θ({b},A = / and Θ(, {a} = Θ(, {b} = Θ(, A = 0. It is routine calculation to check that Θ is transitive and, in fact, is a congruence relation, however a {a} y A Θ({a}, {y} y A a {a} Θ({a}, {y} = ( Θ({a}, {y} Θ({a}, {y} = Θ({a}, {y} = y A but Θ({a},A =. y A Under the additional assumption of commutativity with respect to the usual composition of binary relations, Bakhshi and Borzooei [], stated that the set of all fuzzy congruence relations on a hypergrupoid (H, is a complete lattice. The following example proves that this result is not true even in the crisp case and, thus, it can not be true in a fuzzy framework either. Example. Let H be the set {a,b,c,u 0,u,v 0,v } provided with a commutative hyperoperation which is defined as follows: a a = a b = b b = {a,b}; a c = {u 0,u }; b c = {v 0,v } and x y = {c}, elsewhere Consider R,S : H H {0,} two binary relations, where R is the least equivalence relation containing {(a,b,(u 0,v 0,(u,v } and S the least equivalence relation containing {(a,b,(u 0,v,(u,v 0 }. It is not difficult to check that R and S commute; moreover, easy but tedious calculations show that R and S are compatible with the hyperoperation (i.e. they are congruence relations. However, the only candidate for the meet of R and S, that is, the intersection R S, is not a congruence relation because a(r Sb and for u 0 a c there is no element x b c such that u 0 (R Sx. For the benefit of the reader, a pictorial representation of all the relations involved in this example are shown in Figure. An obvious consequence of the previous counterexample is the convenience of investigating conditions on the nd-groupoid (or hypergroupoid that guarantee the lattice structure on FCon(A. The following result is an immediate consequence from the definition of fuzzy congruence relation. Lemma. Let ρ be a fuzzy congruence relation in an idempotent nd-groupoid (A,. If ρ(a,b > 0 then ab. Moreover, ρ(a,b = ρ(a,c ρ(c,b for all c ab. y A
10 December, 00 : FuzzyCong(IJCM Page of I.P. Cabrera, P. Cordero, G. Gutiérrez, J. Martínez, M. Ojeda-Aciego 0 0 a b a b c R u 0 v v 0 u c R S = S R u 0 v v 0 u Figure. a b a b c c S u 0 v R S Proof Given a, b A such that ρ(a, b > 0, we have that ab because 0 < ρ(a,b ρ(aa,ab = ρ(a,ab Let us now consider c ab. By transitivity ρ(a, b ρ(a, c ρ(c, b. On the other hand, as ρ(a,ab ρ(a,b, we have in particular ρ(a,c ρ(a,b; analogously ρ(ab, b ρ(a, b which implies that ρ(c, b ρ(a, b. Thus As a result, ρ(a,b = ρ(a,c ρ(c,b. ρ(a,c ρ(c,b ρ(a,b Theorem. Let (A, be an nd-groupoid satisfying idempotency and property C, and let ρ be a fuzzy equivalence relation. Then ρ is a congruence relation if and only if the following condition holds: For all a,b,c A with a b we have that ρ(ac,bc ρ(a,b ( Proof The necessity is obvious, thus we will just prove the sufficiency. If ρ(a,b > 0 then, by Lemma., there exists z ab such that ρ(a,b = ρ(a,z ρ(z,b. Property C ensures that a z and b z and then, by Condition ( and symmetry, ρ(ac, zc ρ(a, z and ρ(zc, bc ρ(z, b. Now, by transitivity, ρ(ac,bc X A ( ρ(ac,x ρ(x,bc ρ(ac,zc ρ(zc,bc ρ(a,z ρ(z,b = ρ(a,b u 0 v v 0 u v 0 u
11 December, 00 : FuzzyCong(IJCM 0 Fuzzy congruence relations on nd-groupoids Page 0 of 0 0 From now on, we focus on the search of properties that ensure Condition ( of the previous theorem. Proposition. Let (A, be a commutative, m-associative nd-groupoid satisfying both comparability properties, ρ be a fuzzy congruence relation and a, b, c A. If a b, w ac and z bc with w z then ρ(w,z ρ(a,b. Proof Consider w ac such that wz = z (i.e., w z. By C, we have that a w (w = aw and, by ρ being a fuzzy congruence relation ρ(a,b ρ(aw,bw y bw x aw ρ(x,y = y bw ρ(w, y As a result, it is sufficient to prove that z bw. By using b z (which holds by C and the general hypothesis z bc and w z, and m-associativity, we can write z = bz = b(wz (bwz, that is, z (bwz; therefore, there exists z bw such that z = z z, that is, z z. A similar application of m-associativity, based on the inequalities b z and c w z (which follow from C and transitivity, shows the existence of z bc satisfying z z and therefore z z. Now, recalling that z also belongs to bc by general hypothesis, the comparability property C leads to z = z. As a result of this equality, we have that z z and z z. By commutativity, the relation is antisymmetric and, hence, z = z bw. Recall that the general idea is that, given a b, to prove that ρ(ac,bc ρ(a,b. The previous proposition ensures the inequality ρ(w,z ρ(a,b for elements w ac and z bc such that w z. Now, in order to obtain the inequality for ρ, one has to start from z bc and show the existence of the suitable w ac, and vice versa. Proposition. Let (A, be an m-associative nd-groupoid that satisfies C and, for a,b,c A, consider a b and z bc. Then there exists w ac such that w z. Proof By hypothesis a b and, by C, since z bc, we obtain b z. Now, as the nd-operation is m-associative, the relation is transitive and, therefore, a z. Applying C again on z bc leads to c z; by m-associativity, z = az = a(cz (acz. In particular, we have that z (acz and this implies the existence of w ac such that z = wz, that is, w z. Next, we concentrate on the converse, that is, beginning with an element in ac, find suitable elements in bc so that the congruence holds. This is based on the property of m-distributivity introduced below: Definition. An nd-operation in a set A is said to be m-distributive when, for all a,b,c A, if a b and w ac, then bw bc. It is convenient to remark that m-distributivity arose in the context of multilattices [], although we will not work with this algebraic structure in this paper. Proposition. Let (A, be an m-distributive nd-groupoid that satisfies C and consider a,b,c A. If a b and w ac then there exists z bc such that w z. Proof By m-distributivity, from a b and w ac, we obtain that there exists z bw bc. Now, w z holds by C. Now, we have all the required properties and propositions needed in order to face the main goal of this paper, namely, to prove that under certain circumstances the
12 December, 00 : FuzzyCong(IJCM Page of I.P. Cabrera, P. Cordero, G. Gutiérrez, J. Martínez, M. Ojeda-Aciego 0 0 set of congruences on an nd-groupoid is a complete lattice. Our proof of the complete lattice structure of the set of fuzzy congruences on an nd-groupoid is based on Theorem. and propositions.,. and.. If we summarize all the required hypotheses, we have that the nd-groupoid has to be an m-distributive multisemilattice. Theorem. The set of the fuzzy congruence relations, FCon(M, in an m- distributive multisemilattice M, is a sublattice of FEq(M and, moreover is a complete lattice wrt the fuzzy inclusion ordering. Proof Let {ρ i } i Λ be a set of fuzzy congruence relations in M, consider ρ to be their intersection. From Theorem. we have just to check that, every a,b,c M with a b satisfy that ρ (ac,bc ρ (a,b. From Proposition., if z bc then there exists w ac such that w z and, then, Proposition. implies ρ i (w,z ρ i (a,b for all i Λ. So, ρ (x,z ρ (w,z = ρ i (w,z ρ i (a,b = ρ (a,b x ac w z w z i Λ w z i Λ w ac w ac w ac Analogously, from Proposition. and Proposition., if w ac then there exists z bc such that w z and ρ (w,y ρ (w,z = ρ i (w,z ρ i (a,b = ρ (a,b y bc z w z bc z w i Λ z bc z w i Λ z bc Therefore, ρ (ac,bc ρ (a,b. The proof for the transitive closure of union follows by a routine calculation.. Conclusions and future work Starting with the usual notion of fuzzy congruence relation in a groupoid, we have introduced the definition of fuzzy congruence relation in an nd-groupoid by means of the power extension of the relation to the powerset of the carrier. Our definition is proved to be an adequate generalization of that introduced by Bakhshi and Borzooei in []. Moreover, contrariwise to their claim, we have proved that, if (A, is a hypergroupoid (and thus an nd-groupoid, the set of fuzzy congruences on A, with the usual operations for infimum and supremum is not necessarily a lattice. As a consequence of this negative result, we investigated conditions on the ndgroupoid so that we can guarantee the structure of complete lattice of its set of fuzzy congruences. Such conditions are those of an m-distributive multisemilattice. As future work on this research line, our plan is to keep investigating new or analogue results concerning congruences on generalized algebraic structures, specially in a non-deterministic sense; in this topic, it seems to be important to study the so-called power structures from a universal standpoint [, ]. We will also focus on the corresponding fuzzifications of concepts such as ideal, closure systems and homomorphisms over nd-structures, in the line of [].
13 December, 00 : FuzzyCong(IJCM REFERENCES Page of 0 0 Acknowledgements This paper has been partially supported by projects TIN00--C0-0, TIN00- (Science Ministry of Spain and P0-FQM-00 (Junta de Andalucía. References [] M. Bakhshi and R. A. Borzooei. Lattice structure on fuzzy congruence relations of a hypergroupoid. Inform. Sci., (:0, 00. [] M. Benado. Les ensembles partiellement ordonnés et le théorème de raffinement de Schreier. I. Čehoslovack. Mat. Ž., (:0,. [] I. Bošnjak and R. Madarász. On power structures. Algebra Discrete Math., (:, 00. [] P. Cordero, G. Gutiérrez, J. Martínez, M. Ojeda-Aciego, and I. de las Peñas. Congruence relations on hypergroupoids and nd-groupoids. In th Spanish Conference on Fuzzy Logic and Technology, 00. (To appear. [] P. Cordero, G. Gutiérrez, J. Martínez, M. Ojeda-Aciego, and I. de las Peñas. Congruence relations on multilattices. In th International FLINS Conference on Computational Intelligence in Decision and Control, 00. (To appear. [] P. Das. Lattice of fuzzy congruences in inverse semigroups. Fuzzy Sets and Systems, (:,. [] T. Dutta and B. Biswas. On fuzzy congruence of a near-ring module. Fuzzy Sets and Systems, (:, 000. [] J. Goguen. L-fuzzy sets. J. Math. Anal. Appl., :,. [] R. S. Madarász. Remarks on power structures. Algebra Universalis, (:,. [0] J. Martínez, G. Gutiérrez, I. P. de Guzmán, and P. Cordero. Generalizations of lattices via nondeterministic operators. Discrete Math., (-:0, 00. [] J. Martínez, G. Gutiérrez, I. P. de Guzmán, and P. Cordero. Multilattices via multisemilattices. In Topics in applied and theoretical mathematics and computer science, pages. WSEAS, 00. [] F. Marty. Sur une généralisation de la notion de groupe. In th Congress Math. Scandinaves, pages, Stockholm,. [] J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calviño. Fuzzy logic programming via multilattices. Fuzzy Sets and Systems, (:, 00. [] V. Murali. Fuzzy equivalence relations. Fuzzy Sets and Systems, (:,. [] V. Murali. Fuzzy congruence relations. Fuzzy Sets and Systems, (:,. [] U. M. Swamy and D. V. Raju. Fuzzy ideals and congruences of lattices. Fuzzy Sets and Systems, (:,. [] Y. Tan. Fuzzy congruences on a regular semigroup. Fuzzy Sets and Systems, (:, 00. [] L. A. Zadeh. Fuzzy sets. Information and Control, :,.
Fuzzy congruence relations on nd-groupoids
Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2008 13 17 June 2008. Fuzzy congruence relations on nd-groupoids P. Cordero 1, I.
More informationSets and Motivation for Boolean algebra
SET THEORY Basic concepts Notations Subset Algebra of sets The power set Ordered pairs and Cartesian product Relations on sets Types of relations and their properties Relational matrix and the graph of
More informationIntuitionistic Fuzzy Hyperideals in Intuitionistic Fuzzy Semi-Hypergroups
International Journal of Algebra, Vol. 6, 2012, no. 13, 617-636 Intuitionistic Fuzzy Hyperideals in Intuitionistic Fuzzy Semi-Hypergroups K. S. Abdulmula and A. R. Salleh School of Mathematical Sciences,
More informationQuotient and Homomorphism in Krasner Ternary Hyperrings
International Journal of Mathematical Analysis Vol. 8, 2014, no. 58, 2845-2859 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.410316 Quotient and Homomorphism in Krasner Ternary Hyperrings
More informationKyung Ho Kim, B. Davvaz and Eun Hwan Roh. Received March 5, 2007
Scientiae Mathematicae Japonicae Online, e-2007, 649 656 649 ON HYPER R-SUBGROUPS OF HYPERNEAR-RINGS Kyung Ho Kim, B. Davvaz and Eun Hwan Roh Received March 5, 2007 Abstract. The study of hypernear-rings
More informationChapter 1. Sets and Numbers
Chapter 1. Sets and Numbers 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More informationBoolean Algebras. Chapter 2
Chapter 2 Boolean Algebras Let X be an arbitrary set and let P(X) be the class of all subsets of X (the power set of X). Three natural set-theoretic operations on P(X) are the binary operations of union
More informationEuropean Journal of Combinatorics
European Journal of Combinatorics 31 (2010) 780 789 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Hypergroups and n-ary relations
More informationAxioms of Kleene Algebra
Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.
More informationTree sets. Reinhard Diestel
1 Tree sets Reinhard Diestel Abstract We study an abstract notion of tree structure which generalizes treedecompositions of graphs and matroids. Unlike tree-decompositions, which are too closely linked
More informationClosure operators on sets and algebraic lattices
Closure operators on sets and algebraic lattices Sergiu Rudeanu University of Bucharest Romania Closure operators are abundant in mathematics; here are a few examples. Given an algebraic structure, such
More informationZARISKI TOPOLOGY FOR SECOND SUBHYPERMODULES
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 2018 (554 568) 554 ZARISKI TOPOLOGY FOR SECOND SUBHYPERMODULES Razieh Mahjoob Vahid Ghaffari Department of Mathematics Faculty of Mathematics Statistics
More informationOn the Hypergroups Associated with
Journal of Informatics and Mathematical Sciences Vol. 6, No. 2, pp. 61 76, 2014 ISSN 0975-5748 (online); 0974-875X (print) Published by RGN Publications http://www.rgnpublications.com On the Hypergroups
More informationOn generalized fuzzy sets in ordered LA-semihypergroups
Proceedings of the Estonian Academy of Sciences, 2019, 68, 1, 43 54 https://doi.org/10.3176/proc.2019.1.06 Available online at www.eap.ee/proceedings On generalized fuzzy sets in ordered LA-semihypergroups
More informationFuzzy Sets. Mirko Navara navara/fl/fset printe.pdf February 28, 2019
The notion of fuzzy set. Minimum about classical) sets Fuzzy ets Mirko Navara http://cmp.felk.cvut.cz/ navara/fl/fset printe.pdf February 8, 09 To aviod problems of the set theory, we restrict ourselves
More informationAPPROXIMATIONS IN H v -MODULES. B. Davvaz 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 6, No. 4, pp. 499-505, December 2002 This paper is available online at http://www.math.nthu.edu.tw/tjm/ APPROXIMATIONS IN H v -MODULES B. Davvaz Abstract. In this
More information2MA105 Algebraic Structures I
2MA105 Algebraic Structures I Per-Anders Svensson http://homepage.lnu.se/staff/psvmsi/2ma105.html Lecture 12 Partially Ordered Sets Lattices Bounded Lattices Distributive Lattices Complemented Lattices
More informationFuzzy rank functions in the set of all binary systems
DOI 10.1186/s40064-016-3536-z RESEARCH Open Access Fuzzy rank functions in the set of all binary systems Hee Sik Kim 1, J. Neggers 2 and Keum Sook So 3* *Correspondence: ksso@hallym.ac.kr 3 Department
More informationarxiv: v1 [math.ra] 25 May 2013
Quasigroups and Related Systems 20 (2012), 203 209 Congruences on completely inverse AG -groupoids Wieslaw A. Dudek and Roman S. Gigoń arxiv:1305.6858v1 [math.ra] 25 May 2013 Abstract. By a completely
More informationUniversal Algebra for Logics
Universal Algebra for Logics Joanna GRYGIEL University of Czestochowa Poland j.grygiel@ajd.czest.pl 2005 These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic
More informationPOL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005
POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1
More informationOn fuzzy preordered sets and monotone Galois connections
2015 IEEE Symposium Series on Computational Intelligence On fuzzy preordered sets and monotone Galois connections F. García-Pardo, I.P. Cabrera, P. Cordero, M. Ojeda-Aciego Universidad de Málaga. Andalucía
More informationPrime filters of hyperlattices
DOI: 10.1515/auom-2016-0025 An. Şt. Univ. Ovidius Constanţa Vol. 24(2),2016, 15 26 Prime filters of hyperlattices R. Ameri, M. Amiri-Bideshki, A. B. Saeid and S. Hoskova-Mayerova Abstract The purpose of
More informationSOFT IDEALS IN ORDERED SEMIGROUPS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 58, No. 1, 2017, Pages 85 94 Published online: November 11, 2016 SOFT IDEALS IN ORDERED SEMIGROUPS E. H. HAMOUDA Abstract. The notions of soft left and soft
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationPrime Hyperideal in Multiplicative Ternary Hyperrings
International Journal of Algebra, Vol. 10, 2016, no. 5, 207-219 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6320 Prime Hyperideal in Multiplicative Ternary Hyperrings Md. Salim Department
More informationSOME STRUCTURAL PROPERTIES OF HYPER KS-SEMIGROUPS
italian journal of pure and applied mathematics n. 33 2014 (319 332) 319 SOME STRUCTURAL PROPERTIES OF HYPER KS-SEMIGROUPS Bijan Davvaz Department of Mathematics Yazd University Yazd Iran e-mail: davvaz@yazduni.ac.ir
More informationTopology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:
Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework
More informationNotes on Ordered Sets
Notes on Ordered Sets Mariusz Wodzicki September 10, 2013 1 Vocabulary 1.1 Definitions Definition 1.1 A binary relation on a set S is said to be a partial order if it is reflexive, x x, weakly antisymmetric,
More informationGENERALIZED DIFFERENCE POSETS AND ORTHOALGEBRAS. 0. Introduction
Acta Math. Univ. Comenianae Vol. LXV, 2(1996), pp. 247 279 247 GENERALIZED DIFFERENCE POSETS AND ORTHOALGEBRAS J. HEDLÍKOVÁ and S. PULMANNOVÁ Abstract. A difference on a poset (P, ) is a partial binary
More informationPartial cubes: structures, characterizations, and constructions
Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes
More informationScientiae Mathematicae Japonicae Online, Vol. 4(2001), FUZZY HYPERBCK IDEALS OF HYPERBCK ALGEBRAS Young Bae Jun and Xiao LongXin Received
Scientiae Mathematicae Japonicae Online, Vol. 4(2001), 415 422 415 FUZZY HYPERBCK IDEALS OF HYPERBCK ALGEBRAS Young Bae Jun and Xiao LongXin Received August 7, 2000 Abstract. The fuzzification of the notion
More informationHyperideals and hypersystems in LA-hyperrings
Songklanakarin J. Sci. Technol. 39 (5), 651-657, Sep. - Oct. 017 http://www.sjst.psu.ac.th Original Article Hyperideals and hypersystems in LA-hyperrings Inayatur Rehman 1, Naveed Yaqoob *, and Shah Nawaz
More informationCongruence Boolean Lifting Property
Congruence Boolean Lifting Property George GEORGESCU and Claudia MUREŞAN University of Bucharest Faculty of Mathematics and Computer Science Academiei 14, RO 010014, Bucharest, Romania Emails: georgescu.capreni@yahoo.com;
More informationLinking L-Chu correspondences and completely lattice L-ordered sets
Linking L-Chu correspondences and completely lattice L-ordered sets Ondrej Krídlo 1 and Manuel Ojeda-Aciego 2 1 University of Pavol Jozef Šafárik, Košice, Slovakia 2 Dept. Matemática Aplicada, Univ. Málaga,
More informationG. de Cooman E. E. Kerre Universiteit Gent Vakgroep Toegepaste Wiskunde en Informatica
AMPLE FIELDS G. de Cooman E. E. Kerre Universiteit Gent Vakgroep Toegepaste Wiskunde en Informatica In this paper, we study the notion of an ample or complete field, a special case of the well-known fields
More informationOn the Truth Values of Fuzzy Statements
International Journal of Applied Computational Science and Mathematics. ISSN 2249-3042 Volume 3, Number 1 (2013), pp. 1-6 Research India Publications http://www.ripublication.com/ijacsm.htm On the Truth
More informationBoolean Algebra CHAPTER 15
CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an
More informationPROPERTIES OF HYPERIDEALS IN ORDERED SEMIHYPERGROUPS. Thawhat Changphas. Bijan Davvaz
italian journal of pure and applied mathematics n. 33 2014 (425 432) 425 PROPERTIES OF HYPERIDEALS IN ORDERED SEMIHYPERGROUPS Thawhat Changphas Department of Mathematics Faculty of Science Khon Kaen University
More informationDOI: /auom An. Şt. Univ. Ovidius Constanţa Vol. 25(1),2017, ON BI-ALGEBRAS
DOI: 10.1515/auom-2017-0014 An. Şt. Univ. Ovidius Constanţa Vol. 25(1),2017, 177 194 ON BI-ALGEBRAS Arsham Borumand Saeid, Hee Sik Kim and Akbar Rezaei Abstract In this paper, we introduce a new algebra,
More informationTheory of Computation
Thomas Zeugmann Hokkaido University Laboratory for Algorithmics http://www-alg.ist.hokudai.ac.jp/ thomas/toc/ Lecture 1: Introducing Formal Languages Motivation I This course is about the study of a fascinating
More informationANNIHILATOR IDEALS IN ALMOST SEMILATTICE
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 7(2017), 339-352 DOI: 10.7251/BIMVI1702339R Former BULLETIN
More informationare Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication
7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =
More informationSongklanakarin Journal of Science and Technology SJST R1 Yaqoob. Hyperideals and Hypersystems in LA-hyperrings
Songklanakarin Journal of Science and Technology SJST-0-0.R Yaqoob Hyperideals and Hypersystems in LA-hyperrings Journal: Songklanakarin Journal of Science and Technology Manuscript ID SJST-0-0.R Manuscript
More informationOn the lattice of congruences on a fruitful semigroup
On the lattice of congruences on a fruitful semigroup Department of Mathematics University of Bielsko-Biala POLAND email: rgigon@ath.bielsko.pl or romekgigon@tlen.pl The 54th Summer School on General Algebra
More informationUni-soft hyper BCK-ideals in hyper BCK-algebras.
Research Article http://wwwalliedacademiesorg/journal-applied-mathematics-statistical-applications/ Uni-soft hyper BCK-ideals in hyper BCK-algebras Xiao Long Xin 1*, Jianming Zhan 2, Young Bae Jun 3 1
More informationThe overlap algebra of regular opens
The overlap algebra of regular opens Francesco Ciraulo Giovanni Sambin Abstract Overlap algebras are complete lattices enriched with an extra primitive relation, called overlap. The new notion of overlap
More informationA NATURAL EXTENSION OF THE YOUNG PARTITIONS LATTICE
A NATURAL EXTENSION OF THE YOUNG PARTITIONS LATTICE C. BISI, G. CHIASELOTTI, G. MARINO, P.A. OLIVERIO Abstract. Recently Andrews introduced the concept of signed partition: a signed partition is a finite
More information0 Sets and Induction. Sets
0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set
More informationSup-t-norm and inf-residuum are a single type of relational equations
International Journal of General Systems Vol. 00, No. 00, February 2011, 1 12 Sup-t-norm and inf-residuum are a single type of relational equations Eduard Bartl a, Radim Belohlavek b Department of Computer
More informationChapter 1. Sets and Mappings
Chapter 1. Sets and Mappings 1. Sets A set is considered to be a collection of objects (elements). If A is a set and x is an element of the set A, we say x is a member of A or x belongs to A, and we write
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationMetric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y)
Metric Space Topology (Spring 2016) Selected Homework Solutions HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) d(z, w) d(x, z) + d(y, w) holds for all w, x, y, z X.
More informationLecture Notes: Selected Topics in Discrete Structures. Ulf Nilsson
Lecture Notes: Selected Topics in Discrete Structures Ulf Nilsson Dept of Computer and Information Science Linköping University 581 83 Linköping, Sweden ulfni@ida.liu.se 2004-03-09 Contents Chapter 1.
More informationAn embedding of ChuCors in L-ChuCors
Proceedings of the 10th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2010 27 30 June 2010. An embedding of ChuCors in L-ChuCors Ondrej Krídlo 1,
More informationRESEARCH ARTICLE. On the L-fuzzy generalization of Chu correspondences
International Journal of Computer Mathematics Vol. 00, No. 00, June 2009, 1 11 RESEARCH ARTICLE On the L-fuzzy generalization of Chu correspondences Ondrej Krídlo a and M. Ojeda-Aciego b a Institute of
More informationSome remarks on hyper MV -algebras
Journal of Intelligent & Fuzzy Systems 27 (2014) 2997 3005 DOI:10.3233/IFS-141258 IOS Press 2997 Some remarks on hyper MV -algebras R.A. Borzooei a, W.A. Dudek b,, A. Radfar c and O. Zahiri a a Department
More informationProperties of Boolean Algebras
Phillip James Swansea University December 15, 2008 Plan For Today Boolean Algebras and Order..... Brief Re-cap Order A Boolean algebra is a set A together with the distinguished elements 0 and 1, the binary
More informationL fuzzy ideals in Γ semiring. M. Murali Krishna Rao, B. Vekateswarlu
Annals of Fuzzy Mathematics and Informatics Volume 10, No. 1, (July 2015), pp. 1 16 ISSN: 2093 9310 (print version) ISSN: 2287 6235 (electronic version) http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com
More informationFuzzy Function: Theoretical and Practical Point of View
EUSFLAT-LFA 2011 July 2011 Aix-les-Bains, France Fuzzy Function: Theoretical and Practical Point of View Irina Perfilieva, University of Ostrava, Inst. for Research and Applications of Fuzzy Modeling,
More informationMathematical Reasoning & Proofs
Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
More informationAustin Mohr Math 730 Homework. f(x) = y for some x λ Λ
Austin Mohr Math 730 Homework In the following problems, let Λ be an indexing set and let A and B λ for λ Λ be arbitrary sets. Problem 1B1 ( ) Show A B λ = (A B λ ). λ Λ λ Λ Proof. ( ) x A B λ λ Λ x A
More informationIntroduction to Kleene Algebras
Introduction to Kleene Algebras Riccardo Pucella Basic Notions Seminar December 1, 2005 Introduction to Kleene Algebras p.1 Idempotent Semirings An idempotent semiring is a structure S = (S, +,, 1, 0)
More informationKrivine s Intuitionistic Proof of Classical Completeness (for countable languages)
Krivine s Intuitionistic Proof of Classical Completeness (for countable languages) Berardi Stefano Valentini Silvio Dip. Informatica Dip. Mat. Pura ed Applicata Univ. Torino Univ. Padova c.so Svizzera
More information1. To be a grandfather. Objects of our consideration are people; a person a is associated with a person b if a is a grandfather of b.
20 [161016-1020 ] 3.3 Binary relations In mathematics, as in everyday situations, we often speak about a relationship between objects, which means an idea of two objects being related or associated one
More informationContinuity of partially ordered soft sets via soft Scott topology and soft sobrification A. F. Sayed
Bulletin of Mathematical Sciences and Applications Online: 2014-08-04 ISSN: 2278-9634, Vol. 9, pp 79-88 doi:10.18052/www.scipress.com/bmsa.9.79 2014 SciPress Ltd., Switzerland Continuity of partially ordered
More informationON FUZZY TOPOLOGICAL BCC-ALGEBRAS 1
Discussiones Mathematicae General Algebra and Applications 20 (2000 ) 77 86 ON FUZZY TOPOLOGICAL BCC-ALGEBRAS 1 Wies law A. Dudek Institute of Mathematics Technical University Wybrzeże Wyspiańskiego 27,
More informationIntuitionistic Hesitant Fuzzy Filters in BE-Algebras
Intuitionistic Hesitant Fuzzy Filters in BE-Algebras Hamid Shojaei Department of Mathematics, Payame Noor University, P.O.Box. 19395-3697, Tehran, Iran Email: hshojaei2000@gmail.com Neda shojaei Department
More informationCommutative Algebra Lecture 3: Lattices and Categories (Sept. 13, 2013)
Commutative Algebra Lecture 3: Lattices and Categories (Sept. 13, 2013) Navid Alaei September 17, 2013 1 Lattice Basics There are, in general, two equivalent approaches to defining a lattice; one is rather
More informationPart 1. For any A-module, let M[x] denote the set of all polynomials in x with coefficients in M, that is to say expressions of the form
Commutative Algebra Homework 3 David Nichols Part 1 Exercise 2.6 For any A-module, let M[x] denote the set of all polynomials in x with coefficients in M, that is to say expressions of the form m 0 + m
More information@FMI c Kyung Moon Sa Co.
Annals of Fuzzy Mathematics and Informatics Volume 4, No. 2, October 2012), pp. 365 375 ISSN 2093 9310 http://www.afmi.or.kr @FMI c Kyung Moon Sa Co. http://www.kyungmoon.com On soft int-groups Kenan Kaygisiz
More informationCharacterizations of Regular Semigroups
Appl. Math. Inf. Sci. 8, No. 2, 715-719 (2014) 715 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080230 Characterizations of Regular Semigroups Bingxue
More informationEquational Logic. Chapter Syntax Terms and Term Algebras
Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from
More informationALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.
ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add
More informationarxiv: v1 [math.lo] 30 Aug 2018
arxiv:1808.10324v1 [math.lo] 30 Aug 2018 Real coextensions as a tool for constructing triangular norms Thomas Vetterlein Department of Knowledge-Based Mathematical Systems Johannes Kepler University Linz
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a
More informationIMPLICATION ALGEBRAS
Discussiones Mathematicae General Algebra and Applications 26 (2006 ) 141 153 IMPLICATION ALGEBRAS Ivan Chajda Department of Algebra and Geometry Palacký University of Olomouc Tomkova 40, 779 00 Olomouc,
More informationWhen does a semiring become a residuated lattice?
When does a semiring become a residuated lattice? Ivan Chajda and Helmut Länger arxiv:1809.07646v1 [math.ra] 20 Sep 2018 Abstract It is an easy observation that every residuated lattice is in fact a semiring
More informationSeminaar Abstrakte Wiskunde Seminar in Abstract Mathematics Lecture notes in progress (27 March 2010)
http://math.sun.ac.za/amsc/sam Seminaar Abstrakte Wiskunde Seminar in Abstract Mathematics 2009-2010 Lecture notes in progress (27 March 2010) Contents 2009 Semester I: Elements 5 1. Cartesian product
More informationTWIST PRODUCT DERIVED FROM Γ-SEMIHYPERGROUP
Kragujevac Journal of Mathematics Volume 42(4) (2018), Pages 607 617. TWIST PRODUCT DERIVED FROM Γ-SEMIHYPERGROUP S. OSTADHADI-DEHKORDI Abstract. The aim of this research work is to define a new class
More informationPartially ordered monads and powerset Kleene algebras
Partially ordered monads and powerset Kleene algebras Patrik Eklund 1 and Werner Gähler 2 1 Umeå University, Department of Computing Science, SE-90187 Umeå, Sweden peklund@cs.umu.se 2 Scheibenbergstr.
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationRings and Fields Theorems
Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)
More informationOn the Structure of Rough Approximations
On the Structure of Rough Approximations (Extended Abstract) Jouni Järvinen Turku Centre for Computer Science (TUCS) Lemminkäisenkatu 14 A, FIN-20520 Turku, Finland jjarvine@cs.utu.fi Abstract. We study
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection
More informationON SOME BASIC CONSTRUCTIONS IN CATEGORIES OF QUANTALE-VALUED SUP-LATTICES. 1. Introduction
Math. Appl. 5 (2016, 39 53 DOI: 10.13164/ma.2016.04 ON SOME BASIC CONSTRUCTIONS IN CATEGORIES OF QUANTALE-VALUED SUP-LATTICES RADEK ŠLESINGER Abstract. If the standard concepts of partial-order relation
More informationA n = A N = [ N, N] A n = A 1 = [ 1, 1]. n=1
Math 235: Assignment 1 Solutions 1.1: For n N not zero, let A n = [ n, n] (The closed interval in R containing all real numbers x satisfying n x n). It is easy to see that we have the chain of inclusion
More informationFuzzy Dot Subalgebras and Fuzzy Dot Ideals of B-algebras
Journal of Uncertain Systems Vol.8, No.1, pp.22-30, 2014 Online at: www.jus.org.uk Fuzzy Dot Subalgebras and Fuzzy Dot Ideals of B-algebras Tapan Senapati a,, Monoranjan Bhowmik b, Madhumangal Pal c a
More informationInternational Journal of Mathematical Archive-7(1), 2016, Available online through ISSN
International Journal of Mathematical Archive-7(1), 2016, 200-208 Available online through www.ijma.info ISSN 2229 5046 ON ANTI FUZZY IDEALS OF LATTICES DHANANI S. H.* Department of Mathematics, K. I.
More informationPacket #5: Binary Relations. Applied Discrete Mathematics
Packet #5: Binary Relations Applied Discrete Mathematics Table of Contents Binary Relations Summary Page 1 Binary Relations Examples Page 2 Properties of Relations Page 3 Examples Pages 4-5 Representations
More informationCitation Osaka Journal of Mathematics. 43(2)
TitleIrreducible representations of the Author(s) Kosuda, Masashi Citation Osaka Journal of Mathematics. 43(2) Issue 2006-06 Date Text Version publisher URL http://hdl.handle.net/094/0396 DOI Rights Osaka
More informationOn Regularity of Incline Matrices
International Journal of Algebra, Vol. 5, 2011, no. 19, 909-924 On Regularity of Incline Matrices A. R. Meenakshi and P. Shakila Banu Department of Mathematics Karpagam University Coimbatore-641 021, India
More informationOn weak Γ-(semi)hypergroups
PURE MATHEMATICS RESEARCH ARTICLE On weak Γ-(semi)hypergroups T Zare 1, M Jafarpour 1 * and H Aghabozorgi 1 Received: 29 September 2016 Accepted: 28 December 2016 First Published: 14 February 2017 *Corresponding
More informationIntroduction to generalized topological spaces
@ Applied General Topology c Universidad Politécnica de Valencia Volume 12, no. 1, 2011 pp. 49-66 Introduction to generalized topological spaces Irina Zvina Abstract We introduce the notion of generalized
More informationAn Introduction to the Theory of Lattice
An Introduction to the Theory of Lattice Jinfang Wang Λy Graduate School of Science and Technology, Chiba University 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan May 11, 2006 Λ Fax: 81-43-290-3663.
More informationStrong Deterministic Fuzzy Automata
Volume-5, Issue-6, December-2015 International Journal of Engineering and Management Research Page Number: 77-81 Strong Deterministic Fuzzy Automata A.Jeyanthi 1, B.Stalin 2 1 Faculty, Department of Mathematics,
More informationIntuitionistic L-Fuzzy Rings. By K. Meena & K. V. Thomas Bharata Mata College, Thrikkakara
Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 12 Issue 14 Version 1.0 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals
More informationTopics in Logic, Set Theory and Computability
Topics in Logic, Set Theory and Computability Homework Set #3 Due Friday 4/6 at 3pm (by email or in person at 08-3234) Exercises from Handouts 7-C-2 7-E-6 7-E-7(a) 8-A-4 8-A-9(a) 8-B-2 8-C-2(a,b,c) 8-D-4(a)
More informationFACTOR CONGRUENCES IN SEMILATTICES
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 52, Número 1, 2011, Páginas 1 10 FACTOR CONGRUENCES IN SEMILATTICES PEDRO SÁNCHEZ TERRAF Presented by R. Cignoli Abstract. We characterize factor congruences
More informationSome Properties of a Set-valued Homomorphism on Modules
2012, TextRoad Publication ISSN 2090-4304 Journal of Basic and Applied Scientific Research www.textroad.com Some Properties of a Set-valued Homomorphism on Modules S.B. Hosseini 1, M. Saberifar 2 1 Department
More information